Test configurations for K-stability and geodesic rays

  • Phong D
  • Sturm J
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Abstract

Let $X$ be a compact complex manifold, $L\to X$ an ample line bundle over $X$, and ${\cal H}$ the space of all positively curved metrics on $L$. We show that a pair $(h_0,T)$ consisting of a point $h_0\in {\cal H}$ and a test configuration $T=({\cal L}\to {\cal X}\to {\bf C})$, canonically determines a weak geodesic ray $R(h_0,T)$ in ${\cal H}$ which emanates from $h_0$. Thus a test configuration behaves like a vector field on the space of K\"ahler potentials ${\cal H}$. We prove that $R$ is non-trivial if the ${\bf C}^\times$ action on $X_0$, the central fiber of $\cal X$, is non-trivial. The ray $R$ is obtained as limit of smooth geodesic rays $R_k\subset{\cal H}_k$, where ${\cal H}_k\subset{\cal H}$ is the subspace of Bergman metrics.

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APA

Phong, D. H., & Sturm, J. (2007). Test configurations for K-stability and geodesic rays. Journal of Symplectic Geometry, 5(2), 221–247. https://doi.org/10.4310/jsg.2007.v5.n2.a3

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